Problem: Find one value of $x$ that is a solution to the equation: $(4x+1)^2+9(4x+1)=-18$ $x=$
Solution: We could solve for $x$ by expanding $(4x+1)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Let's look at the given equation: $({4x+1})^2+9({4x+1})=-18$ If we let ${p}={4x+1}$, we can see that this equation is in the form: ${p}^2+9{p}=-18$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2+9{p}&=-18\\\\ {p}^2+9{p}+18&=0\\\\ ({p}+6)({p}+3)&=0\\\\ {p}=-6\ &\text{or} \ \ {p}=-3 \end{aligned}$ Since ${p}={4x+1}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${4x+1}=-6\ \ \ \text{or} \ \ \ {4x+1}=-3$ When we solve ${4x+1}=-6$, we find that $x=-\dfrac{7}{4}$. When we solve ${4x+1}=-3$, we find that $x=-1$. In conclusion, the two solutions of the equation $(4x+1)^2+9(4x+1)=-18$ are $x=-\dfrac{7}{4}$ and $x=-1$. [Is there another way to solve for x?]